Abstract : For ordinary differential equations it is well known that an equilibrium solution is stable if the linearized equation has only exponentially decaying solutions. The differential-difference equation u' (t+s) - u' (t) + au (t + )s) = g (t, u (t), u (t+ s)), a > O, s > O, when linearized, also has decaying exponentials exp (st), but the characteristic roots s approach the imaginary axis. The problem arises -- do solutions of this equation decay. An affirmative answer is obtained for this problem, the rate of decay depending upon the smoothness of the initial data and nonlinear term and the rate of approach of the characteristic roots to the imaginary axis. Then this result is applied to a transmission line problem. Also, an example is given of a homogeneous linear ordinary differential-difference equation with constant coefficients, whose characteristic roots all lie in the left half plane, having an unbounded solution. (Author)